A Pluripotential-Theoretic Approach to Finiteness of Polarized Calabi--Yau Manifolds

Source abstract

We propose a pluripotential-theoretic framework for studying finiteness questions for polarized Calabi--Yau manifolds. The central idea is to use global $m$-Hessian capacities as a quantitative intermediate between complex potential theory and global geometric boundedness. We formulate a chain of uniform estimates connecting $m$-Hessian capacity, volume--capacity inequalities, capacity--perimeter estimates, Sobolev bounds, and non-collapsing of the associated Ricci-flat Kähler metrics. Under suitable regularity assumptions, these estimates lead to uniform diameter and curvature control and hence to metric compactness. We then describe the algebraic part of the argument. Uniform projective embeddings with bounded degree lead, through the Macaulay--Gotzmann theory, to only finitely many possible Hilbert polynomials and hence to a finite collection of Hilbert schemes. Ehresmann's fibration theorem then converts boundedness of the smooth algebraic families into finiteness of diffeomorphism and topological types. The resulting framework isolates the main analytic and algebraic bottlenecks in a potential-theoretic approach to Calabi--Yau finiteness. In particular, the capacity--perimeter estimate and the derivation of uniform algebraic embedding data from the preceding metric estimates constitute the essential steps requiring further development. Thus the paper provides a structured program linking complex Hessian potential theory with geometric and algebraic boundedness, rather than assuming that these implications are automatic.